Lemma 104.3.2. With assumptions and notation as in Cohomology of Stacks, Lemma 103.15.1. We have
on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).
Lemma 104.3.2. With assumptions and notation as in Cohomology of Stacks, Lemma 103.15.1. We have
on unbounded derived categories (both for the case of modules and for the case of abelian sheaves).
Proof. Let \tau = {\acute{e}tale} (resp. \tau = fppf). Let \mathcal{F} be an abelian sheaf on \mathcal{X}_\tau . By Cohomology of Stacks, Lemma 103.15.3 the canonical (base change) map
is an isomorphism. The rest of the proof is formal. Since cohomology of abelian groups and sheaves of modules agree we also conclude that g^{-1} Rf_*\mathcal{F} = Rf'_* (g')^{-1}\mathcal{F} when \mathcal{F} is a sheaf of modules on \mathcal{X}_\tau .
Next we show that for \mathcal{G} (either sheaf of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{flat,fppf}) the canonical map
is an isomorphism. To see this it is enough to prove for any injective sheaf \mathcal{I} on \mathcal{X}_\tau the induced map
is an isomorphism for all n \in \mathbf{Z}. (Hom's taken in suitable derived categories.) By the adjointness of f^{-1} and Rf_*, the adjointness of Lg_! and g^{-1}, and their “primed” versions this follows from the isomorphism g^{-1} Rf_*\mathcal{I} \to Rf'_* (g')^{-1}\mathcal{I} proved above.
In the case of a bounded complex \mathcal{G}^\bullet (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}) the canonical map
is an isomorphism as follows from the case of a sheaf by the usual arguments involving truncations and the fact that the functors L(g')_!(f')^{-1} and f^{-1}Lg_! are exact functors of triangulated categories.
Suppose that \mathcal{G}^\bullet is a bounded above complex (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}). The canonical map (104.3.2.1) is an isomorphism because we can use the stupid truncations \sigma _{\geq -n} (see Homology, Section 12.15) to write \mathcal{G}^\bullet as a colimit \mathcal{G}^\bullet = \mathop{\mathrm{colim}}\nolimits \mathcal{G}_ n^\bullet of bounded complexes. This gives a distinguished triangle
and each of the functors L(g')_!, (f')^{-1}, f^{-1}, Lg_! commutes with direct sums (of complexes).
If \mathcal{G}^\bullet is an arbitrary complex (of modules or abelian groups) on \mathcal{Y}_{lisse,{\acute{e}tale}} (resp. \mathcal{Y}_{fppf}) then we use the canonical truncations \tau _{\leq n} (see Homology, Section 12.15) to write \mathcal{G}^\bullet as a colimit of bounded above complexes and we repeat the argument of the paragraph above.
Finally, by the adjointness of f^{-1} and Rf_*, the adjointness of Lg_! and g^{-1}, and their “primed” versions we conclude that the first identity of the lemma follows from the second in full generality. \square
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